Markus Hanke

13 hours ago, DanMP said:

So, here, on the Earth surface, we also see the light coming from stars straight above us "blueshifting" as long as our accelerometer shows that we are "accelerating" upward?

Technically yes, there will be some amount of frequency shift, though in practice the effect is quite small for a weak field such as the Earth’s.

13 hours ago, geordief said:

Do you have any other (or complementary) intuitive ways  of  sitting this process in our pattern of thoughts?

That’s hard to answer, since whether something is considered intuitive or not depends on the person. I kind of like the paths lengths way of looking at it, since most people can relate to it.

8 hours ago, externo said:

Moreover, the path taken by the twin is longer and not shorter, what is shorter is its proper time.

Total accumulated proper time equals the geometric length of the path through spacetime, as I’ve mentioned already. The crucial point is that the two paths are not of equal lengths.

8 hours ago, externo said:

Yet, Wikipedia says that there is a paradox except in the case where we postulate a privileged reference frame.

I think you should read the article more carefully.

8 hours ago, externo said:

How do you know that there is no preferred reference frame where the light is isotropic? You can't know.

I said there’s no rest frame to light, so it makes no sense to speak of “speed relative to waves”. 

8 hours ago, externo said:

There is no gravitational field in an accelerated frame of reference, we are in flat space-time,

There’s no tidal gravity in an accelerated frame, meaning that

\[R_{\mu\nu}=0; W{^{\mu}}{_{\nu\alpha\beta}}=0\]

and therefore

\[R{^{\mu}}{_{\nu\alpha\beta}}=0\]

However, there is a homogenous gravitational field due to proper acceleration locally, according to the equivalence principle. The Riemann tensor vanishes for such homogenous fields, so spacetime remains of course flat as expected. The metric in the accelerating frame, which now contains a term which is equivalent to a gravitational potential, is isomorphic to the Minkowski metric, also as expected.

20 hours ago, externo said:

So you don't have any explanation, you just stick to the math.

The explanation is exactly what the maths says - pick a different path, and you’ll walk a different distance. There’s nothing else to it.

20 hours ago, externo said:

You used Lorentz mathematics, they are the ones who mathematically resolved the paradox.

No, I showed you that there’s no paradox that needs resolving. It seems to me that you’re wilfully refusing to “get” this. 

20 hours ago, externo said:

Because it is he who causes this increase in the signal by changing its speed relative to the waves.

There’s no such thing as “relative to waves”, because light has no rest frame. The speed is always between emitter and receiver.

20 hours ago, externo said:

And why does the turning twin not need to wait for the signal emitted from Earth to propagate to him before observing the blueshift?

Because it’s he who experiences acceleration locally in his frame. The equivalence principle tells us that uniform acceleration is locally equivalent to a uniform gravitational field; differently put, the accelerated twin sits at a different gravitational potential, which implies frequency shift.

21 hours ago, externo said:

If it is physical it implies that the Earth suddenly ages for real during the U-turn

I never mentioned any U-turn - I made it explicitly clear that I made no assumptions about what the path actually looks like, other than it being light-like (thus differentiable everywhere). Why? Because that’s irrelevant, since the difference in clock readings only depends on the total lengths of the two paths. It’s a global measure along the entire journey. Others here have repeatedly pointed this out too. And since both path length and proper acceleration are invariant measures, both twins agree on the outcome.

21 hours ago, externo said:

So math agrees with Lorentz but not with Einstein.

That’s a meaningless statement. I used Einsteinian SR to show how the twin scenario requires no “resolution”.

21 hours ago, externo said:

the acceleration period can be reduced to an infinitely short time

Really? How do you physically realise an instantaneous U-turn with infinite acceleration?

Once again, the clock times are integral measures along the entire journey.

21 hours ago, externo said:

If the the frequency of the electromagnetic signal increases, it is because the one accelerating has changed its velocity in relation to the signal

In relation to the emitter, not the signal. There’s no rest frame for light.

17 hours ago, externo said:

Inertia comes from the ether. When we accelerate the ether manifests itself as inertia.

You didn’t answer my question, you’re just making another unsubstantiated claim.

17 hours ago, externo said:

Instead of the speed of light changing, it is simultaneity changing and the speed of light remaining constant.

It’s invariant, not constant. There’s a difference here. But yes, since the speed of light is finite and the frames are separated, there’s of course relativity of simultaneity; and since acceleration is a change of velocity, the relationship between the frames is time-dependent.

17 hours ago, externo said:

The article clearly establishes that there are physical discontinuities in Minkowski space

I’ve already pointed out that the metric of a relativistically rotating disk is not the Minkowski metric. 

17 hours ago, externo said:

You talk about paths in Minkowski spacetime as if it had a physical reality.

SR is of course “just” a mathematical model, same as any other model in physics. However, the lengths of paths through spacetime correspond to what clocks physically display, so that’s pretty “real” to me.

Like I said, if you connect the same two points in any territory along different paths, it’s hardly a surprise that, in general, these come out at different lengths. What deeper mechanism do you need for that?

17 hours ago, externo said:

You see that the twin ages less over the duration of the entiere journey, which is Lorentz's interpretation.

That’s no one’s ‘interpretation’, it’s what the standard SR maths say, as I have shown you earlier. There are no interpretations involved in this.

17 hours ago, externo said:

As soon as it leaves the Earth's frame of reference it begins to age less, so its perception of symmetry during the inertial journey is false. He believes that the Earth is aging less than him when in reality it is the opposite.

Only one of the frames experiences proper acceleration, which is not a relative measure - both frames agree on who’s accelerated and who isn’t. So there’s not any symmetry in this situation, except during those times when both frames are inertial. But that symmetry concerns the instantaneous rate at which the clocks tick, not their readings - if one of the clocks has first undergone non-inertial motion, and then becomes inertial, their tick rates are symmetrical, but nonetheless one clock displays a different total proper time. The effect of non-inertial motion is accumulative, which is what I pointed out above with line integrals - you have to account for the entire journey.

17 hours ago, externo said:

This is not consistent with Einstein's interpretation that the traveler ages less during interial travels.

I don’t know what you mean by this. 

9 hours ago, externo said:

Redshift is the manifestation of a change in speed between light and the accelerating one.

No it isn’t - frequency shift is due to the fact that energy is a frame-dependent quantity, it has nothing to do with changes in c. Light propagates at the same speed in both frames, but they don’t agree on its energy.

14 hours ago, externo said:

The ether is detectable during acceleration

Really? Please provide references to peer-reviewed experiments that unambiguously (ie not just in your “interpretation”) detect the ether. What is it made of? What are its equations of motion?

14 hours ago, externo said:

Einstein's SR is not capable of processing accelerations, otherwise explain to me what happens during an acceleration according to Einstein. The speed of light remains constant and time changes its simultaneity, is that the explanation?

You really need to stop repeating things that have already been shown to be wrong. You’re not doing yourself any favours.

What exactly do you want explained? One of the frames measures acceleration (using a local accelerometer), and the frames are related by the transformations given in my link, instead of Lorentz transformations.

14 hours ago, externo said:

This article also explains some problems with SR:

This concerns the rotation of rigid objects at relativistic speeds - it’s been known for a long time that this involves a metric that isn’t Minkowski, so that’s hardly a “problem with SR”, but falls outside its scope.

14 hours ago, externo said:

They say that the traveling twin has traveled the greatest spatial distance and the shortest temporal distance.

No. These integrals concern total accumulated proper time; this is an invariant quantity that’s not relative to anything. I deliberately did not use relative quantities, but invariant line integrals.

What the equations say is that (in this sign convention) it is always the inertial clock that accumulates the most proper time between a given pair of events in spacetime. IOW, any clock that doesn’t trace out a geodesic between these events will record less proper time in comparison - and we know of course that there’s only one such geodesic for any given pair of events in Minkowski spacetime.

Therefore, there’s no paradox, and nothing needs resolving. It’s simply that, if you choose two different paths, you can’t in general expect them to be of equal lengths. 

14 hours ago, externo said:

When exactly does he get younger in relation to Earth?

The dilation between clocks is an integral measure - it concerns a comparison between total geometric lengths of world lines, so one must take into account the entire journey. Thus in general you can’t reduce this to a single instant. The most we can say is that the accumulated times begin to diverge the instant the travelling clock ceases to be at rest relative to the Earth-bound clock. Also, he never gets “younger” - he just ages less.

So again - SR very much does resolve this, contrary to your claim.

PS. I remind you again to bear in mind what the twin scenario is fundamentally about - it’s a comparison between total accumulated proper times on two clocks that connect the same two events along different paths. And this is precisely what I mathematically described, not more and not less. 

On 4/18/2024 at 4:28 PM, externo said:

You cannot resolve the paradox with Einstein's SR

The discussion seems to be going off on a lot of tangents now, so I propose we bring it back to the above original claim. Here’s how I would approach this personally:

The twin scenario concerns two clocks (E-Earth, T-Travelling) which start together at rest at some event A; twin T then separates for a period and eventually they reunite again at rest at another event B.

In SR, the total accumulated time on a clock travelling between a pair of events is defined to equal the geometric length of the worldline traced out by that clock through spacetime:

\[\tau=\int_{C}ds\]

where C(t,x) is the path taken. Since in SR we have

\[ds=\sqrt{\eta_{\mu \nu } dx^{\mu}dx^{\nu}}\]

we get, for the Earth-bound twin (no change in spatial location!):

\[\tau_{E}=\int_{A}^{B}\sqrt{\eta_{\mu \nu } dx^{\mu}dx^{\nu}}=\int_{A}^{B}\sqrt{\eta_{tt}}=c\int_{A}^{B}dt\]

For the travelling twin, on the other hand, we get:

\[\tau_{T}=\int_{C}\sqrt{\eta_{\mu \nu } dx^{\mu}dx^{\nu}}=\int_{C}\sqrt{c^2dt^2-dx^2-dy^2-dz^2}\]

wherein we don’t assume any specifics about the path C, other than that it connects A and B, and remains light-like everywhere.

I invite you to verify yourself that

\[\tau_{T} < \tau_{E}\]

by choosing any light-like path C and running the numbers in these line integrals.

What this means is that, in SR, an inertial clock will always trace out the longest path through spacetime between events, ie it will accumulate the most time. Any path deviating from being a geodesic will necessarily be shorter (or vice versa, depending on your sign convention). Since the travelling twin starts out at rest wrt to its partner, but then travels away, it cannot remain on a geodesic - thus SR guarantees that its clock accumulates less time than the Earth-bound twin. Far from being a paradox, it is a natural consequence of the geometry of spacetime.

Note that I made no assumptions here about the specific form of the path C, only that it is light-like and runs through A and B; thus it is realisable by any test particle with mass.

So yes, SR very much can resolve this, contrary to your claim.

It should also be explicitly noted that the total time dilation does not depend on the magnitude of acceleration at any one instant, but only on the total path length of the world line.

Lastly, an experimental test of these predictions can be made by comparing the decay rates of unstable particles in accelerators against stationary reference samples. See eg Bailey et al (1977) with regards to CERNs muon storage ring.

19 hours ago, externo said:

You are talking about mathematics

I was talking about something you claimed - that SR cannot handle acceleration. This is manifestly wrong, because it very much can.

19 hours ago, externo said:

I talk about the physical interpretation of these equations

What interpretations? The ether in LET is by design undetectable, and its presence has no physical consequences, thus both SR and LET make identical predictions. This isn’t a matter of interpretation, but belief - do you choose to believe and assume the presence of an entity that cannot be detected, has no physical consequences, and is not actually needed to obtain the correct dynamics?

19 hours ago, externo said:

There is no doubt that Lorentz's interpretation works because it uses classical kinematics, but Einstein's interpretation uses a anomalous kinematics that must be proven consistent with the physical world.

Not only has SR been exhaustively tested over the past ~120 years and thus shown to be consistent with the physical world, but we also use it directly in many aspects of modern technology. A lot of parts of the very machine you are using to make your posts here have been designed based directly or indirectly on SR, and it evidently works well.

Also, SR kinematics are classical, just like Newtonian kinematics are.

13 hours ago, externo said:

But there is quaternionic spacetime where time is the scalar dimension of space

But APS is just based on a subgroup of the full Cl(1,3) Clifford spacetime algebra…? It is a convenient and very useful formalism, but hardly new physics.

6 hours ago, externo said:

I have already cited this one :

RG is a social network, not a peer review journal. Some stuff there is useful, but one has to use caution.

Personally, I stopped reading when the author first talked about twins in Minkowski spacetime, and then gave a scenario as example where one twin is in orbit around a star. Not an especially convincing paper, and the author clearly has an agenda.

15 hours ago, externo said:

Einstein's SR does not know how to deal with accelerations.

This is a misconception which is as common as it false. SR is a model of Minkowski spacetime - it describes the relationship between any set of frames within this paradigm, irrespective of what their states of relative motion and acceleration are. In the special case of inertial motion, this relationship is simply a hyperbolic rotation in spacetime (=Lorentz transformation); if acceleration is involved, the relationship is a little more complicated, but nonetheless well defined:

https://en.m.wikipedia.org/wiki/Acceleration_(special_relativity)

There’s no “paradox” in the twins scenario that somehow needs resolution, it’s simply a straightforward consequence of the geometry of Minkowski spacetime, which has to do with the lengths of world-lines.

12 hours ago, Maartenn100 said:

Since Einstein there is no universe with absolute time properties (and absolute space properties). 

Given that you assume the theory of relativity to be valid, this will imply that an absence of time would also mean an absence of gravity between bodies. Clearly, this is not what we observe.

On 4/5/2024 at 6:53 AM, Mordred said:

Agreed, on that I tend to think more on global distributions lol

I understand

Just as a side remark though, it is interesting to note that the EFE also admits cosmological vacuum solutions. An example is the Kasner metric; this kind of universe is completely empty (no energy-momentum and no cosmological constant), yet still not Riemann-flat in general, and metrically expands in an anisotropic manner. One can contrast this against the case of FLRW, and finds that it is precisely the presence of matter/radiation that enables an isotropic expansion.

Obviously this is purely academic, but nonetheless instructive.

16 hours ago, dimreepr said:

but I know that he will acknowledge that he had to be taught the basics

Of course  I was very fortunate that I grew up in a part of the world were an excellent basic education is freely available to everyone. I consider myself lucky.

23 hours ago, Mordred said:

point taken in so far as the Schwartzchild is a static spherically symmetric mass solution lol

I was referring to the exterior metric, which has T=0 everywhere. The main point I am trying to make is that the EFE is a purely local constraint on the metric - but one must also account for sources that are distant in space and/or time, which happens via boundary conditions when solving the equations. To put it differently, the metric isn’t in general determined solely by local energy-momentum alone, but also by distant sources (and cosmological constant).

9 hours ago, Mordred said:

For example spacetime without particles to generate a mass term has zero curvature.

Careful here - the field equations emit many vacuum solutions without the presence of particles. For example, the standard Schwarzschild solution represents a spacetime that is everywhere vacuum (T=0 at all points), yet not flat.

In general terms, the EFE states a local (!) equivalence between the Ricci tensor and a source term - but a vanishing Ricci tensor does not necessarily imply a flat manifold.

16 hours ago, Maartenn100 said:

Valid conclusion: Every observation of space is relative. THUS ALSO the observation of expanding space, as an interpretation of the observation of the Doppler effect of light emitted from distant galaxies, by 'expansion of space' moving away from us.

I think you’re forgetting that these distant objects are not just moving away from us, but also from each other. Metric expansion happens for all distances, not just those centred on Earth (which would be exceedingly weird). Furthermore, there’s a precise relationship between distance and recession velocity, irrespective of which direction you look.

Be careful to place to much stock in a single observation - you have to remember that cosmological redshift is not the only data point available to us; there’s quite a number of different observations involved, and the Lambda-CDM model is currently (!) the one that best fits all the available data. It’s not just based on redshift.

14 hours ago, TheVat said:

I hope it is clear I was speaking of support and developing coping skills to further life goals and not a "cure" for neurodivergence.

Yes, don’t worry 👍

14 hours ago, TheVat said:

My training pointed me towards looking around at the environment and seeing if stimulation could be reduced so as to ease the felt need for it

Yes, exactly.

The key point is this - the most severe forms of autism aside for the moment, most autistics are perfectly happy and do not suffer if they are allowed to “live in their own world”, ie structure their daily lives according to their own needs, and have the ability to pursue their chosen interests. It is chiefly when we’re forced into environments, situations and pursuits that are fundamentally at odds with our psycho-neurological makeup that we get into distress, which triggers things like the need to stim, sensory overload, and autistic burnout.

I think one possible way to help the autistic community at large, would be to exist within a system that provides some form of UBI (universal basic income). This would remove the pressure of having to somehow earn a basic living, while providing opportunities to contribute to society according to our strengths and abilities - and these contributions cannot always be measured in monetary terms. Needless to say, this would also benefit other groups whose contributions to society at large can’t be measured in money.

Just my two cents

11 hours ago, MSC said:

That said, who decides what neurotypical is? Where is the baseline or control group of humanity that we can point to as having "normal" brains? 

Yes, that’s a good point. I think there’s no objective measure for this - normal is what accords to the general consensus of what constitutes normality, so it’s down to who is in the majority. It’s a numbers game.

9 hours ago, MSC said:

I know I sound like a broken record when talking about contextualism, but context really is one of the most important factors in discussing everything.

Yes, your absolutely right. As they say: judge a fish by its ability to climb trees, and it will never amount to anything. That’s often how it is with ND people. 

14 hours ago, TheVat said:

There were persons I worked with who faced such obstacles and yes, welcomed a path towards surmounting them. 

Yes, absolutely. But remember the context of this discussion - the claim was made that autism is a disease that’s due to blood toxicity, and can be cured on that basis. This of course is utter nonsense, and as an autistic person myself I’d really wish there was a way to rid us of such snake oil salesmen (and there are many of them).

Autism - like other forms of neurodivergence - isn’t a “disease”, it’s a difference in brain physiology that has genetic, developmental and environmental factors involved. It cannot be “cured” in the classical sense, for that reason. It manifests across a range of areas - social, cognitive, executive functioning etc -, and impacts a person’s quality of life anywhere on a spectrum from very mildly to extremely severely. Succinctly stated, neurodivergent people tend to have support needs - we must live in a world that is fundamentally designed for neurotypicals (both culturally and evolutionary), but because we aren’t neurotypical, some expectations can be hard for us to meet and some situations difficult to handle, and we need support and accommodations to manage them. Some of us need only a little (or no) support and accommodation, others need a lot, and for some it’s debilitating, and they need 24/7 support and care. 

So in my personal opinion, it’s not about “curing” autism - which fundamentally suggests that we’re somehow deficient and need fixing, which is a questionable stance to hold. Rather, it’s about recognising that autistics may need extra supports and accommodations in certain areas, and being willing to offer those. There are also meaningful interventions available for at least some of the more severe and debilitating manifestations of autism, and I completely agree that these should be offered so long as the aim is to improve quality of life, and not just to make people “less autistic” and “more normal”. That’s an important difference. We will never be neurotypical, but with the right help we can become less dependent on external supports.

There’s another thing I’d like to mention, which is not so often talked about - some commonly held life goals and values that are normal and generally unquestioned in the neurotypical world may not be shared by all autistics. For example, wanting and needing to be social and around others, wanting to acquire possessions and material goods, wanting to have family and procreate, conforming to gender norms, following generally accepted blue prints for how life should be lived, ideas around what has value and what doesn’t, concepts of what gives us meaning and joy in life etc. While this is of course very individually different, not all of us autistics share these goals and values, so we end up in a situation where we are forced to try and fit into a society and culture that feels fundamentally alien to us. We can’t be our real selves, but must train ourselves to wear a certain mask and act a role so that we might appear to be able to meet the expectations of a neurotypical society, simply because it’s practically and logistically very difficult to exist outside that system. We feel like we don’t have a choice, so we live a life that’s at odds with who we are. This creates a lot of suffering and struggle. So at least part of our suffering isn’t due to autism itself, but due to demands and expectations placed on us by others to “be a certain way”. 

15 hours ago, TheVat said:

flicking their fingers back and forth in front of their faces for hours

Such behaviour - called stimming - causes us no suffering. On the contrary, it feels soothing and comfortable, and dissipates the perceived pressure of sensory overload. I do it by (gently) pressing certain places on my hands for example, and it helps me to self-regulate. Who are others to say that this is wrong and needs fixing?

I would suggest that the problem here isn’t this particular behaviour pattern itself, but how it is perceived by others. From a neurotypical point of few it appears meaningless, odd and not normal, and it is tacitly assumed that the one engaged in it does it only because he feels compelled to do so, and thus suffers. But it’s not like that - it’s a self-regulation tool, like people take a painkiller when they have a headache. Both help increase well-being. The problem is only that society has deemed taking painkillers to be acceptable behaviour, but not flicking your fingers. If you stop an autistic person from stimming, or shame them into hiding the behaviour, you aren’t acting to promote their overall welfare, even if as a result they might appear “more normal” and supposedly fit in better. I would suggest that training autistics to appear less autistic is generally not in our overall best interest, unless we ourselves specifically ask for such interventions.

This is a difficult ethical question, particularly for severely autistic children - there’s ways and training methods to make them appear less autistic, so they can function better in neurotypical society. The price they pay of course is that they’re forced to be something they’re not, that they’re forced to play a part in a story they themselves haven’t read or understood. Are you really doing them a favour, are they really suffering less afterwards? Most people in the autistic community who underwent such childhood interventions seem to say that no, it wasn’t in their best interest, even if it did enable them to function better in society. The general consensus is that providing supports and accommodations on an individual basis if and when needed, is probably the best way for most of us. Sometimes that means we need a lot of support, and that’s the measure of a modern enlightened society - how it relates to their weakest members, who perhaps can’t contribute in traditional ways.

Caveat: if the behaviour is dangerous, or injurious to one self or others, as sadly sometimes is the case, then of course intervention is necessary regardless.

16 hours ago, exchemist said:

I would not have guessed, Markus.

You might have, had you met me in person  Though I generally mask quite well, so even IRL it may not be immediately obvious to people.

16 hours ago, exchemist said:

Except that you are rather good with hard mathematical physics…..🙂

Autistic people very often develop intense interests in certain topics, and sometimes hyperfocus so much on those that they become quite knowledgeable and proficient in them, even though they aren’t formally trained experts in their chosen field. It should be noted though that - contrary to the commonly imagined stereotype - this is not always in an STEM related subject. It can be anything from art, history, music, politics etc, to something very specific like trains, jeans, or Elvis Presley. One may also have a hyper-interest in more than one thing at a time, and sometimes those interests can change over time. We are as different from one another as any other group of people, and the idea that all autistics are math geniuses is a common but inaccurate misconception.

On another note, autism is not the same as savant syndrom - these are distinct diagnoses.

On 3/24/2024 at 12:37 AM, Guille Yacante said:

The solution is hemodialysis.

The solution is for people who are not themselves autistic, and who evidently don’t understand what autism even means, to stop proposing “solutions”.

I am autistic, and I am not a problem that needs to be solved.

7 hours ago, KJW said:

However, in two dimensions, there is only one type of curvature; in three dimensions, there are two types of curvature; and in four or more dimensions, there are three types of curvature.

I’m probably forgetting something obvious but…what are you referring to by “types”? Do you mean the Weyl tensor, Ricci tensor, and Ricci scalar?

Not the AT specifically, but I’ve done a few of the long-distance trails around Europe. Meant to do the Continental Divide Trail a few years back, but had to cancel due to Covid.

Agree with everything @TheVat said, especially the bits about water. One can’t be careful enough, there’s nothing more miserable than a stomach bug on trail. Personally I’m partial to my good old Sawyer Squeeze filter, has served me well for many a trail!

11 hours ago, TheVat said:

Unclear to me what "any background geometry" would be here. 

This would be contributions from distant sources, as opposed to local ones, as well as contributions from any non-zero cosmological constant. Basically anything that stops spacetime from being completely flat before you account for any local energy-momentum.

11 hours ago, TheVat said:

As @Genady pointed out, causation can be defined operationally, e.g.  I push two distant planets together and local curvature increases in "response" to greater local mass.

I agree, in this type of scenario you have clear causation in an operational sense. However, I was really thinking more of an isolated system where all parts remain in free fall at all times. Energy-momentum is locally conserved (the divergence of the tensor vanishes) - but then so is curvature (Einstein tensor). You cannot locally create nor destroy Einstein curvature, any more than you can create or destroy energy-momentum. You can only shift these around, and have them change form - so which ‘causes’ which?

11 hours ago, TheVat said:

The stress-energy tensor includes mass, right?

Not directly, but it contains energy density.

18 hours ago, Genady said:

In spite of this, a 'causative relationship' exists operationally, i.e., we can, in principle, manipulate the source and this would affect geometry, but there is no way to manipulate geometry without manipulating the source. The 'symmetry' is broken.

2 hours ago, KJW said:

One can't dismiss a causative relationship between two quantities simply because the mathematical relation between them is an equality. One really needs to examine the logical aspect of the relationship to establish whether there is a causative relationship and in which direction it occurs.

While these comments are certainly true, I think the relationship isn’t as trivial as it might appear; after all, a vanishing Einstein tensor doesn’t necessarily imply a flat spacetime, so these equations form only a local constraint on geometry, but they don’t uniquely determine it. Any background geometry is as much considered to be a ‘source’ as is local energy-momentum, when it comes to working out the particular form of a metric at a certain event. Furthermore you have the non-linearities of the constraint itself, which, in some sense, might also be considered a ‘source’. But those contributions of course don’t explicitly appear.

Personally, I just think of spacetime as pure geometry - the only difference between vacuum and non-vacuum is how the Riemann tensor decomposes (Weyl and Ricci curvature), so I envision it purely geometrically all the way. In that way of thinking, no question of causation arises, you just have ever-shifting geometries.

But maybe that’s just weird old me

14 hours ago, joigus said:

Mass to GR is nowhere near anything like charge is to Yang-Mills theory (our paradigm of an honest-to-goodness QFT field theory).

That’s because mass never appears in the GR field equations - what is generally called the source term here is the energy-momentum tensor. One must also remember what these equations actually say - they state a local equivalence between a certain combination of components of the Riemann tensor (the Einstein tensor) and the energy-momentum tensor. Nowhere does it claim a ‘causative relationship’, but instead it says that these two are the same thing (up to a constant of course); neither one comes first and ‘causes’ the other.

11 hours ago, KJW said:

You and I discussed this in the thread about the cosmological expansion of time as well as space.

Indeed. Just wanted to mention this again quickly in case other readers find it helpful to have some geometric intuition what the various aspects of curvature - Riemann, Weyl, Ricci - actually mean.

6 hours ago, KJW said:

This means that the Weyl conformal tensor is zero, and therefore according to my way of looking at this, there is no gravity (although there is still the curvature associated the Ricci tensor and scalar).

Another way to say this is that, in this kind of spacetime and under geodesic motion, shapes (ie angles) are preserved, whereas volumes and surfaces are not.